Simplify the following expression and state the condition under which the simplification is valid. $q = \dfrac{t^3 - 5t^2 - 6t}{4t^3 + 12t^2 - 216t}$
Solution: First factor out the greatest common factors in the numerator and in the denominator. $ q = \dfrac {t(t^2 - 5t - 6)} {4t(t^2 + 3t - 54)} $ $ q = \dfrac{t}{4t} \cdot \dfrac{t^2 - 5t - 6}{t^2 + 3t - 54} $ Simplify: $ q = \dfrac{1}{4} \cdot \dfrac{t^2 - 5t - 6}{t^2 + 3t - 54}$ Since we are dividing by $t$ , we must remember that $t \neq 0$ Next factor the numerator and denominator. $ q = \dfrac{1}{4} \cdot \dfrac{(t - 6)(t + 1)}{(t - 6)(t + 9)}$ Assuming $t \neq 6$ , we can cancel the $t - 6$ $ q = \dfrac{1}{4} \cdot \dfrac{t + 1}{t + 9}$ Therefore: $ q = \dfrac{ t + 1 }{ 4(t + 9)}$, $t \neq 6$, $t \neq 0$